Generalized Adjoint Method

TL;DR

This paper introduces a generalized adjoint method using CR-calculus to compute gradients in optimization problems involving complex, non-holomorphic functions, extending the applicability of the adjoint method to electromagnetism and signal processing.

Contribution

It presents a novel generalized adjoint method based on CR-calculus that handles non-holomorphic functions in complex variables, broadening the scope of gradient computation techniques.

Findings

Enables gradient computation for non-holomorphic functions

Applicable to inverse problems in electromagnetism and signal processing

Maintains efficiency of the adjoint method in complex variable contexts

Abstract

The adjoint method is an efficient way to numerically compute gradients in optimization problems with constraints, but is only formulated to differentiable cost and constraint functions on real variables. With the introduction of complex variables, which occur often in many inverse problems in electromagnetism and signal processing problems, both the cost and constraint can become non-holomorphic and hence non-differentiable in the standard definitions. Using the notion of CR-calculus, a generalized adjoint method is introduced that can compute the direction of steepest ascent for the cost function while enforcing the constraint even if both are non-holomorphic.